Devices and techniques for logical processing

ABSTRACT

The invention is directed to apparatus, methods, systems and computer programs which permit a simplification or reduction of the description of a complex digital circuit. This is accomplished by using a completely new system of propositional logic ( 13 B), representing the logic of a logical circuit to be designed as points and vectors in a vector space, simplifying the logic so represented to a simpler form using vector transformations, and designing the circuit using the simpler form. The invention is also directed to apparatus, methods, systems and devices which will implement the vector logic system in optical forms, including free-space optical methods, flat-optical methods, colorimetric methods, and methods using a polarization-based AND-gate.

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority from and is related to U.S.provisional application Serial No. 60/238,007, filed Oct. 6, 2000,Attorney Docket No. 52254-014 entitled: OPTICAL AND GATE AND SWITCHINGDEVICE USING POLARIZATION PHOTOCHROMISM. The contents of thatprovisional are hereby incorporated by reference in their entirety.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] This invention relates to the field of logical processing and,more particularly to devices and techniques for implementing and forsimplifying digital logic.

[0004] The invention relates to the field of logical processing, todevices and techniques for simplifying two- and multi-level digitallogic, using the proposed vector form of the digital logic, and todevices and techniques for implementing the said vector logic in theform of optical processors, circuits composed of optical logic gates orswitches, including therefore both optical multiplexers and opticaldemultiplexers, as well as integrated optical circuits.

[0005] 2. Description of Related Art

[0006] Logic can be described as techniques and operations by which onemoves from what one knows to be true to new truths. The principles oflogic have been applied in the design and operation of digital logiccircuits. Modern-day computers and other processing devices haveutilized digital logic extensively. Many of the problems to whichdigital logic can be applied are complex, involving many independentvariables. This results in extremely complex logical circuits in whichlarge numbers of operations are performed. The cost associated withmanufacturing and fabrication of such complex digital circuits is great.It would be highly desirable to reduce the size of these circuits whilepreserving the same functionality, and thereby to reduce themanufacturing cost of their fabrication and to improve their performanceand speed.

[0007] A number of optically active materials are known in the art.Among these are photochromic polymers as described in an article byKunihuro Ichimura, “Photochromic Polymers”, in a text by John C. Cranoand Robert J. Gugilelmetti, eds., entitled Organic Photochromic andThermochromic Compounds, Vol. 2, Kluwer, New York, 1999, pp. 9-63, thecontents of which are incorporated herein by reference thereto.

[0008] Digital computers are of course well known, but more recentlyoptical computers have been developed which can perform logicalfunctions using optical elements. These logical processors can inprinciple perform logical switching functions as fast as is physicallypossible, and there is also no expensive optical-electric-optical(“OEO”) conversion process required to link them with present-dayoptical telecommunications systems. To date, however, they have not beenfeasible largely on account of lack of scalability and the absence offully scalable AND- and NAND- and other logic gates. The inventiondisclosed here makes up this lack.

SUMMARY OF THE INVENTION

[0009] It is an object of this invention to provide new digital 2-leveland multi-level simplification methods using vector logic.

[0010] It is another object of this invention to provide plans forall-optical processors and circuits using vector logic.

[0011] It is yet another object of this invention to provide plans foroptical Multiplexers, Demultiplexers, flip-flops, AND gates and othersimilar devices

[0012] The invention is directed to apparatus, methods, systems andcomputer program products which permit a simplification of the logicrequired for performing a certain function to a minimum set of logicalelements of operations. This permits the complexity of digital circuitryto be reduced and the speed of the computation to be increasedcorrespondingly. It also improves reliability and diminishes materialcosts.

[0013] The logical application is accomplished using a system ofpropositional logic in which propositions are represented as vectors ordisplacements in a space. This is applied to the simplification problem,the problem of finding a method for reducing logical schema to ashortest equivalent. Specifically, an exemplary novel feature of thepresent invention includes placement of an origin within a Boolean cube,and thus making the Boolean cube into a vector space in which vectorscan be translated, added and multiplied, rather than a static structureof bit representations in 1s and 0s.

[0014] The techniques given here are applied to the simplification orminimization problem, the problem of finding a method for reducinglogical schemata to their shortest equivalents, in both two-level andmulti-level forms. Applications of the system of vector logic toproblems of electronic circuit minimization, to free-space opticalprocessing, to flat-optical processing, to logical processing usingcolor images, and to the design of logic gates, including the AND-gateand NAND-gates, using a polarization implementation of the vector logicare described. The polarization implementation is fully scalable as itrequires only four passive elements: beamsplitters, reflectors,polarizers and retarders.

[0015] The present invention provides various exemplary methods forimplementing the vector principles for propositional logic for opticalcomputation. The first exemplary implementation is flat-optical, whereina Mach-Zehnder interferometer-like device is used to operate in eitherSOP or POS-form, which is the basic “cell” in the Karnaugh map sense foran all-optical processor. AND-, XOR-, XNOR- and NAND-gates may beconstructed from these cells using the vector optical implementation. Asecond exemplary implementation includes a system wherein the logic isimplemented in sequences of spatial light modifiers. In a thirdexemplary implementation of the present invention, basic principles ofcolorimetry are used for a simple colorimetric optical processingsystem. A fourth exemplary implementation of the present inventionincludes a Mach-Zehnder “cell” used for a device which does not dependon a bistable ‘and’ sum sigmoid filter.

[0016] In general, in one aspect, the invention features a novel logicsystem.

[0017] In another aspect, the invention features an opticalimplementation of the novel logic system to produce all types of opticallogic circuits.

[0018] Additional advantages of the present invention will becomeapparent to those skilled in the art from the following detaileddescription of exemplary embodiments of the present invention. Theinvention itself, together with further objects and advantages, can bebetter understood by reference to the following detailed description andthe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] The objects, features and advantages of the system of the presentinvention will be apparent from the following description in which:

[0020]FIG. 1 represents two-dimensional space for propositions.

[0021]FIG. 2 illustrates the propositions p and p v q. in the space ofFIG. 1.

[0022]FIG. 3 is a diagram of the vector two-dimensional space showingthe conjunctional normal schemata normal schemata or CNS-plane.

[0023]FIG. 4 is a diagram of the vector two-dimensional space showingthe alternational normal schemata or the ANS-plane.

[0024]FIG. 5 is a diagram showing modus ponens in the CNS-plane.

[0025]FIG. 6 is a diagram showing modus tollens in the CNS-plane FIG. 7is a diagram showing the disjunctive syllogism in the CNS-plane.

[0026]FIG. 8 is the diagram showing how the ANS- and CNS-planes relate.

[0027]FIG. 9 is a diagram illustrating a simplifying operation withinthe ANS-space.

[0028]FIG. 10 shows an extension of the ANS-plane of FIG. 4 to athree-dimensional ANS-space.

[0029]FIG. 11 shows an extension of the CNS-space to three dimensionstogether with a hypothetical syllogism.

[0030]FIG. 12A illustrates a hypothetical syllogism with three variablesin the CNS-space.

[0031]FIG. 12B shows a view of the hypothetical syllogism in thethree-dimensional CNS-space.

[0032]FIG. 13A illustrates a cancellation technique used in simplifyinglogical representations and in accordance with the invention.

[0033]FIG. 13B shows the representations of FIG. 13A in graphical form.

[0034]FIG. 13C illustrates implication and equivalence.

[0035]FIGS. 14A, 14B and 14C illustrate a solution to the simplificationproblem using the techniques of the invention.

[0036]FIG. 15 shows a 4-clause schema simplified.

[0037]FIG. 16A shows a 3-clause schema simplified.

[0038]FIG. 16B shows the truth-table for the representation of FIG. 16A.

[0039]FIG. 17 shows a 4-variable vector diagram simplification.

[0040]FIG. 18A is a diagram illustrating the Fix Rule for d=2.

[0041]FIG. 18B is an illustration of an example of the Fix Rule.

[0042]FIG. 19 is an illustration of the Fix Rule for d=3.

[0043]FIG. 20 illustrates application of the invention to situations inwhich developed normal formulas are not the point of departure.

[0044]FIG. 21 illustrates the equivalence of a developed alternationalnormal form and its undeveloped counterpart.

[0045]FIG. 22 illustrates the simplification of an undeveloped set ofstatements.

[0046]FIG. 23 illustrates another simplification of an undeveloped setof statement taken from Quine.

[0047]FIG. 24 illustrates an equivalence within the set of statementsshown in FIG. 23.

[0048]FIG. 25 illustrates the Consensus Theorem.

[0049]FIG. 26 illustrates the dual of the Consensus Theorem.

[0050]FIG. 27 illustrates a superfluity shown in FIG. 23.

[0051]FIG. 28 illustrates a target circuit to be simplified inaccordance with the invention.

[0052]FIG. 29 shows a simplest circuit equivalent to the target circuit.

[0053]FIG. 30 is an illustration of optical computation of modus ponens.

[0054]FIG. 31 is an illustration of interferometric processing for modusponens.

[0055]FIG. 32 illustrates an optical element used for disconjunction andconjunction in a free-space optical processing.

[0056]FIG. 33 is an illustration of flat optical processing.

[0057]FIG. 34 is an exemplary optical AND-Gate in accordance with thepresent invention.

[0058]FIG. 35 illustrates the Sigmoid Characteristic for an AND-Filterin accordance with the present invention.

[0059]FIG. 36 is an exemplary optical XOR-Gate in accordance with thepresent invention.

[0060]FIG. 37 is an exemplary optical XNOR-Gate in accordance with thepresent invention.

[0061]FIG. 38 is vector representation of an exemplary optical NAND-Gatein accordance with the present invention.

[0062]FIG. 39 is an exemplary optical NAND-Gate in accordance with thepresent invention.

[0063]FIG. 40 illustrates an exemplary implementation of an exemplaryCNS-cell in accordance with the present invention.

[0064]FIG. 41 is a chart of vector logical equivalents of “If p then q”.

[0065]FIG. 42 shows a computation of modus ponens using SLMimplementation.

[0066]FIG. 43 is a view of an SLM device embodying the functionalityshown in FIG. 42.

[0067]FIG. 44 is an illustration of the same vector addition utilizingsequences of SLMs.

[0068]FIG. 45 is an illustration of calorimetric computation of modusponens.

[0069]FIG. 46 is a colorimetric simplification of pq v p{overscore (q)}.

[0070]FIG. 47 shows the colorimetric vector space.

[0071]FIG. 48 is an illustration of the simple colorimetric sum device.

[0072]FIG. 49 is an elaboration of the device shown in FIG. 48.

[0073]FIG. 50 shows the wavelength bands of the filter for the deviceshown in FIG. 49.

[0074]FIG. 51 is a 3-D AND-cell using polarization and vector positionfor optical computation.

[0075]FIG. 52 is the hookup of the device in FIG. 51 to the next inputgate.

[0076]FIG. 53 is a COIN (coincidence) cell based on FIG. 51.

[0077]FIG. 54 is a half-added based on FIG. 51.

[0078]FIG. 55 is a MUX based on FIG. 51.

[0079]FIG. 56 is a (pNANDq)NAND(pNANDq) gate for p and q input based nFIG. 51.

[0080]FIG. 57 is the same, for p and −q input.

[0081]FIG. 58 is an exemplary Polarization NOR-gate.

[0082]FIG. 59 is an exemplary NOR-gate for −p-q input.

[0083]FIG. 60 is the same for p-q input.

[0084]FIG. 61 is the same for −pq input.

[0085]FIG. 62 is the same for −p-q input.

[0086]FIG. 63 is the schematic for a wholly conservative logicpolarization corridor for the AND-function.

[0087]FIG. 64 shows a Conjunctional Normal Form Truth-Table.

[0088]FIG. 65 is a MUX using the principle of FIG. 63.

[0089]FIG. 66 is a flip-flop for all possible paths using the principlesof FIG. 63.

[0090]FIG. 67 is the same, for S=1, R=0, present Q=1.

[0091]FIG. 68 is a spatial path-coding for p, q, −p, −q in thepropagation of rays through a short fiber.

[0092]FIG. 69 illustrates an optical AND gate and switching device usingpolarization photochromism.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0093] In the following description, for the purposes of explanation,numerous specific details are set forth in order to provide a morethorough understanding of the present invention. It will be apparent,however, to one skilled in the art that the present invention may bepracticed without these specific details.

[0094] Part I describes a system of propositional logic in whichpropositions are represented as vectors or displacements in a space.Part II gives the application of the system to the simplificationproblem, the problem of finding a method for reducing a truth-functionalschemata in alternational normal form to a shortest equivalent. Part mis about applications:

[0095] (i) to problems of electrical circuit minimization; (ii) tofree-space optical processing; (iii) to “flat” optical processing; (iv)to logical processing using colorimetry; and (v) to polarization basedprocessing.

Part I

[0096] Imagine a space in which the co-ordinates from the origin 0 arepropositional addresses or possibilities, for example as depicted inFIG. 1. Let (1, 0) be the propositional address p, and (0,1) thepropositional address q. Then (1, 1) is the point p, q, and we can letthe sign “+” be an operation on p and q which is defined by distance anddirection from the origin, by which ‘p+q+p+q’ is an instruction to gotwo units in a p-ward direction, and two units in a q-ward direction.The operation performed by someone obeying this instruction iscommutative and associative.

[0097] We can now represent the proposition that p as a directedline-segment or vector along the p- or x-axis from the origin 0 to thepoint or propositional address p in the space, representing the vector pin boldface as is standardly done to distinguish it from the possibilityp which is represented as the point at the arrowhead of p. We can alsolet the vector q be the proposition q, represented in the space as thevector pointing straight up the q- or y-axis to the point q.

[0098] Now if we build up the space interpreting the operation “+” as“v”, the x- and y-axes will obviously represent lines of logicalequivalence. At (2, 0), or p, p, for example, we will find the arrowheadof p v p, and at (3, 0) the arrowhead of p v p v p. These and the restof the proposition vectors along the p-axis are logically equivalent tothe base vector p. At (0, 2), or q, q, we will find the arrowhead of q vq. Thus p and q will stand in for the usual unit vectors i and j. (Thevector space of propositions can however have infinitely many directionssuch as s, t, u, v . . . , which will become important later on when atechnique is given to simplify propositions with large numbers ofliterals.)

[0099] We are also now in a position to represent the proposition p v qin the space as p+q, the vector resultant of the vectors p and q, whichtravels from the origin to p, q. Then p v q is itself a vector.

[0100] I will call a vector diagram for the propositional calculus suchas FIG. 2 a V-Diagram for the proposition or schema. The V-diagram canbe further built up by adding the negation symbols for the negativevectors p and q in the negative or reverse directions along theirrespective axes. So we arrive at all of the literals, which are singleletters and negations of single letters, and we can also find pairs ofsingle negated or unnegated letters, the propositions p v {overscore(q)}, and {overscore (p)} v {overscore (q)}.

[0101] Let us call the vector two-space in FIG. 3 the CNS-plane for theplane of the “conjunctional normal schemata”.

[0102] We are now free to explore another plane, for example asillustrated in FIG. 4, the plane of the alternational normal schemata,or the ANS-plane, in which the points are not alternations butconjunctions, and the vector operation “+” within the space isinterpreted as alternation. The ANS-plane and the CNS-plane are duals,so that each point in each plane correspond to its dual in the otherplane. This also means that the uniting operation in the CNS-plane isrelated to the dual of the operation in the ANS-plane, and vice versa.In the CNS-plane “+” is alternation, and so in the ANS-plane it isconjunction. The operation “1” in “α→β” in the CNS-plane is to be readas implication or the assertion of the conditional. In the ANS-plane “1”is to be read as the denial of the negation of implication, which is thedenial of the conjunction of the antecedent with the negation of theconsequent. The whole ANS-plane is to be read as a systematic set ofdenials, the denials that the propositions given at the base of thevector arrowheads imply a contradiction. This will be obvious if weremember that arrows ending at the origin rather than those issuing fromit, as in the CNS-plane, are assertions in the ANS-plane.

[0103] In both of the planes certain familiar truths appear asexpressions of the main principle which governs “+” or vector addition,the so-called parallelogram law of Galileo. In the CNS-plane we canthink of the premises of an argument as component vectors, and theresultant as the conclusion. Then an elementary valid argument-form inthe CNS-plane is a parallelogram starting at the origin 0 in that plane.The conjunction of the alternations yields the conclusion, and we getmodus ponens appearing as in Diagram 5. If the vectors are representedas displacements around 0 in the V-diagram, the modus ponens in theCNS-plane is the set of displacements p q Premise 1 −1 1 + {overscore(p)} v q Premise 2 1 0 p Conclusion = 0 1 q

[0104] Modus tollens, as illustrated in Diagram 6, appears as: p qPremise 1 −1 1 + {overscore (p)} v q Premise 2 0 −1 {overscore (q)}Conclusion = −1 0 {overscore (p)}

[0105] The disjunctive syllogism, as illustrated in Diagram 7, appears,with its displacement matrix, as: p q Premise 1 1 1 + p v q Premise 2 −10 {overscore (p)} Conclusion = 0 1 q

[0106] Consider now the relation between the CNS-plane and theANS-plane. There clearly is one, as they share the literals and theall-important origin 0. The two planes can be brought into harmony if werepresent them, arbitrarily, as lying above and below the origin in aspace whose third dimension runs along the conjunction-alternation axis,putting alternation at the top and conjunction at the bottom.

[0107] The result is a space, or the part of it near 0, with two planesabove and below the origin. The origin 0 appears in the vertical axisbetween the two planes. The whole space of FIG. 8 generates furtherprinciples of the propositional calculus. Take p v q in the top righthand corner. Negating it comprehensively, in all three dimensions, ordeveloping it through the origin, gives the point {overscore (p)}{overscore (q)} in the ANS-plane. This is one of the two forms ofDeMorgan's theorem. Its other form can be found by comprehensivelynegating pq in the ANS-plane, and travelling through 0 to {overscore(p)} v {overscore (q)} in the CNS-plane. The CNS-/ANS-space as a wholehas an intriguing and beautiful structure, as it combines the dimensionsof alternation and conjunction, the various propositions formed fromatomic p and q, and the dimension of negation.

[0108] Operations within the ANS-space have “+” representingconjunction. When all the non-equivalent conjunction points areestablished in the space, pairs and other combinations of the givenpoints or conjunctions are given as alternations or vectors. So we get aresultant of pp from pq v p{overscore (q)} by relating the two vectorsto the origin 0 in a parallelogram (FIG. 9).

[0109] In the CNS-space, on the other hand, the corresponding operationproduces alternations, and the operation within the space which combinesthem is conjunction. So we get sets of conjunctions, e.g. thoseimportant ones involving {overscore (p)} v q, which are among the moreimportant arguments of natural deduction.

[0110] Assume now in the ANS-space a third proposition r, and a thirddimension z in which the unit vector r is to be found. So we get ther-plane, the one swept out by the vector r. This space can also berepresented in two dimensions on the page. In FIG. 10 thenegation-affirmation axis, which follows the z-axis in the order ofrotation of the variables about 0, is inserted to prevent the occlusionof lines and points.

[0111] To check the validity of the hypothetical syllogism (−1, 1, 0)(0, −1, 1), (−1, 0, 1), with three variables, in the CNS-space, we canrepresent it as in FIG. 11.

[0112] As illustrated below, and in FIGS. 12A and 12B, the of thevalidity argument appears, using three sets of coordinates, as p q rPremise 1 −1 1 0 + {overscore (p)} v q Premise 2 0 −1 1 {overscore (q)}v r Conclusion = −1 0 1 {overscore (p)} v r

[0113] Note the simplicity of the given representation or perspective onthe hypothetical syllogism in FIG. 11, matched only by the simplicity ofthe pqr string −1, 1, 0, 0, 1-, 1, −1, 0 1, which is merely a set ofinstructions for displacements in a 3-space.

[0114] The vector system can be used in the CNS-space to display otherprinciples, for example implications, by which {overscore (p)} v qimplies p→q. It also shows that {overscore (p)} v q implies 0→{overscore(p)} v q, as well as {overscore (q)}→{overscore (p)} and {overscore (p)}v {overscore (q)}→0. Furthermore, the vector system shows nicely theprinciple of material equivalence, which states that (p→q)(q→p) isequivalent to p

q (FIG. 13).

[0115] The starting point of all these vectors, together with thedirection, gives the end point. “Together with” here means treating thepoints and directions algebraically as themselves directions from theorigin. This yields a cancellation technique in which a starting-pointof 0 is cancellation of no literal, and an end point of 0 is thecancellation of all the literals Starting End Point Direction Point 0{overscore (p)} v q {overscore (p)} v q p {overscore (p)} v q q{overscore (q)} {overscore (p)} v q {overscore (p)} p v {overscore (q)}{overscore (p)} v q 0

[0116] Parallel principles can be given for the ANS-plane. Here “α→β”means the same as in the CNS-plane; which is α⊃β, the conditional, butthe reason is hard to see, though interesting. Take the proposition p→qin the CNS-plane. It is represented by (among others) an arrow from thepoint p to the point q. In the ANS-plane we find an arrow from p to q.Call it v. But what does v mean? Note that the CNS-plane vector from pto q is true if v is false. For v is p{overscore (q)}, and the samevector or direction as p{overscore (q)}→0. If we want the ANS-planevectors to represent truth, we must read them as the denials of theconjunction of the proposition p at the base of the arrow with thenegation of the proposition q at the arrowhead, or −(p{overscore (q)}).Each arrow in the ANS-plane then reliably represents a conditional.

[0117] This reveals something further about the all-important 0, theorigin. We have just learned that in the ANS-plane an arrow from p to qis p{overscore (q)}, to be read however as a negation. So what does α→Omean? O is p{overscore (p)}. So the conjunction of α and −(O) of aα−(pp). But this is α({overscore (p)} v p), which is equivalent to α.

[0118] Similarly, in the CNS-plane, all the arrows which depart from 0represent an instance of α v β, where α is 0. Take an arrow from 0 to pv q. 0 is the tautology p v {overscore (p)}. The negation of thisp{overscore (p)}, and so the arrow to the point p v q is −(p v{overscore (p)}) v pq. But the first disjunct of this is equivalent top{overscore (p)}, and so it is always false. Hence the alternation isequivalent to the second disjunct, or the assertion pq.

[0119] If a vector in the ANS-space is directed towards 0, 0 has theeffect of reversing the truth-values of the base propositions. Movingtowards 0 from the base (p,q) in the CNS-space we get the vector p v q.0 has the effect of putting p and q through the Sheffer-function “|”.The vector moving away from 0 in the CNS-space towards e.g. (p,q) isalso the vector from the base (p,q) to 0, and so it is the vector{double overscore (p)} v {double overscore (q)} or p v q.

[0120] In a dual fashion, if we are moving towards 0 in the ANS-space,we get the base values, so that pq→0 is pq. From 0, a vector to({overscore (p)},{overscore (q)}) will thus be pq. In the ANS-space 0has the effect of putting p and q through the dagger function “↓”, bywhich p↓q is {overscore (p)}{overscore (q)}.¹

Part II

[0121] The simplification problem is the problem of reducingtruth-functional schemata (or, in the system I am describing, systems ofvectors in the ANS-space) to their shortest equivalents. A practicalmethod for doing this, in alternational normal form continues, as Quineobserves (Quine, 1982, p. 78), to be suprisingly elusive.

[0122] In the ANS-space “vector logic” can be applied to the problem inthe following way. Take the schema pq v p{overscore (q)}, which as wellas implying p is equivalent to p. To simplify it, form the parallelogramfrom the origin 0, pq and p{overscore (q)} to the resultant or vectorsum point. Call it

, for “implicant”. The vector acting at

, which is in this case pp, implies pq v p{overscore (q)}. So

splits up alternationally, into its components, pq v p{overscore (q)},towards the origin.

[0123] Next note that pq is equivalent to pqp, so that the arrowhead atpq can be dragged to pqp. But pp can also be dragged to p. Now we havean arrow from p to pqp. But this arrow an be translated into a positionon top of the arrow from p{overscore (q)} to pp. The same procedureyields a double-headed arrow between pq and pp, and the result can beread as pq v p{overscore (q)}

p.

[0124] When an implicant splits up into its alternations towards theorigin, if there is a proposition σ(for “simplest equivalent”) at thecenter of the parallelogram formed by 0, the disjuncts of a two-clausetarget schema, and

, then σ is a shortest equivalent of the target schema. But this onlyworks for pairs of schemata which do have an

-point.

[0125] The general simplification procedure, in the ANS-space, is asfollows.

[0126] (1) Represent the alternational normal schema, the target schemat, as a set of vectors in the ANS-space. Each clause or disjunct of t isa position vector (i.e. one pointing to 0) with 0 at one corner of aparallelogram made of propositional addresses to the

-point at the other. Any two other outside vertices of such aparallelogram are implicants

which are among the original clauses of t.

[0127] (2) Pick any two clauses. If there is a propositional address σat the midpoint between the component clauses, the vector from

to σ, i.e. σ, is the simplification of and can replace the relevantclauses of t, as in the case where t is pq v pq,

is pp and σ is p.

[0128] (3) Generate

-implicants until each clause or vector has been used at least once. Ifa disjunct d of t cannot be used because it forms no propositionaladdress with any other disjunct, then d must appear unmodified in thefinal schema which is the simplification of t.

[0129] (4) If an

-point exists in t, delete the vectors which produce it in favor of thevector from

to 0.

[0130] (5) For a clause in a schema which subsumes another clause, e.g.pqr v pq, eliminate the subsuming clause, in this case pqr, leaving pq.Implications arising from subsumption can be written into the wholevector system of t as components where relevant. For example, an arrowcan be drawn from pqr to pq in the above example.

[0131] Rule (5) applies for example to pq v p, which is an undevelopedor unbalanced schema in which pq subsumes p. How does pq v p simplify top, when it seems to yield p v q? The Answer, which cashes the metaphorof “subsumption”, is that p really represents a plane, in a 3-space,sweeping out the whole p-domain, or any ANS-schema with p in it. So itis a kind of type fallacy to represent pq alongside p in a single schemaas if they were to be treated separately. For pq, and p{overscore (q)},are really “elements” of p itself. A cube is not so many faces and somany lines, but it can be represented as lines producing faces or viceversa. As a matter of philosophy, therefore, vector logic can availitself, as rule (5) does, of a preliminary use on Quine's operation (i)from “A Way to Simplify Truth Functions”, which has us ‘drop thesubsuming clause . . . if one of the clauses of alternation subsumesanother . . . ’. Quine's operation (i) also replaces α v {overscore(α)}φ with α v φ, and the same for the corresponding α-schemata (Quine,1955, p. 627).

[0132] (6) Couples such as pq v {overscore (p)}{overscore (q)} or{overscore (p)}{overscore (q)}s v pq{overscore (s)} cannot be summed tozero, the origin.

[0133] (7) Translate vectors as in FIG. 14. Any superpositions ofparallel arrows in opposite directions represent equivalences. (a) Dropthe longer clause at the end of any double-headed arrow. (b) Drop pairs,triples etc. of double-headed arrows which meet at a point in favor ofthe vector from that point to 0. (c) Drop a vector or clause in thetarget schema which is itself the resultant of any other two vectors.

[0134] (8) A simplification is complete if in the system which replacesthe target schema: (a) no vectors or clauses are subsumed by others (seeRule 5); (b) no double-headed vectors remain, or, in other words, if allequivalences in the system have been exploited.

[0135] Take next the simplification of the four-clause target schema pqrv pq{overscore (r)} v {overscore (p)}qr v p{overscore (q)}{overscore(r)}. The first job is to plot the target schema in a V-diagram. We gettwo parallelograms, with two i-points, qrqr and p{overscore(r)}p{overscore (r)}, and two σ-points, qr and p{overscore (r)}, whichare final in the sense that they do not generate a further σ-point.Hence the target schema is equivalent to qr v p{overscore (r)}.

[0136] Now take the simple-looking three-clause schema pqr vpq{overscore (r)} v p{overscore (q)}{overscore (r)}. The resultant is pqv p{overscore (r)}.

[0137] Here the σ- and

-points function as before. But something else has happened. The vectorpq{overscore (r)} has been used twice, once along with pqr to give pq,and again, with p{overscore (q)}{overscore (r)}, to give p{overscore(r)}. Why was pq{overscore (r)} not exhausted by its first use, and whycan it be used again? The Answer can be seen by looking at thetruth-table for pqr v pq{overscore (r)} v p{overscore (q)}{overscore(r)}, which is 1. pqr T 2. pq{overscore (r)} T 3. p{overscore (q)}r 4.p{overscore (q)}{overscore (r)} T 5. {overscore (p)}qr 6. {overscore(p)}q{overscore (r)} 7. {overscore (p)}{overscore (q)}r 8. {overscore(p)}{overscore (qr)}

[0138] Truth, it could be said, is not exhausted by use. The p{overscore(r)} of line 2 is so to speak redundant, as line 2 has already beencaptured by the disjunct pq, and so line 4 has had half of its workalready done.

[0139] This simplification procedure is theoretically an improvement onthe techniques used in Karnaugh maps (Garrod and Borns, 1991, p. 153ff.), as it needs no wrapping around and can be used mechanically andeasily on more than four variables—any number fits into the “propositioncircuit”, which gradually turns from a square, with two variables, intoa hexagon, with three, and finally into a circle, with an infinitenumber of variables. With four variables, the logical space is as givenin FIG. 17.

[0140] The whole figure in FIG. 17 is a “measure polytope” or hypercube,though one with a further complex internal structure. There is nolimitation of tessellation to the number of propositional variables orvectors p, q, r s . . . that can be handled, because the space isderived not from a closed figure, such as a cube, but from a sheaf oflines in the geometrical sense. Not all closed figures tessellate. Allthe lines of the multi-dimensional sheaves are coincident.

[0141] Consider in FIG. 17 a simplification from pqr{overscore (s)} vpq{overscore (r)}{overscore (s)} v pq{overscore (r)}s v {overscore(p)}qr{overscore (s)} v p{overscore (q)}{overscore (r)}s v {overscore(p)}q{overscore (r)}{overscore (s)} to p{overscore (r)}s v q{overscore(s)}. This proceeds as shown, with the six vectors reducing to two. Thefirst vector or Φ-point in the simplification, p{overscore (r)}s,results from the implicand pair pq{overscore (r)}s v p{overscore (qr)}s,dropping the up-down q/{overscore (q)} component. In this case we have asimplification from two four-letter schemata to one three-letter schema.The remaining four schemata are all needed to fix the Φ-pointq{overscore (s)}. Both pairs pqr{overscore (s)} v p{overscore(q)}{overscore (r)}s and pq{overscore (r)}{overscore (s)} v {overscore(p)}qr{overscore (s)} must give a fix on the same Φ-point if thereduction from four literals to two is to be justified. For either pairby themselves is not sufficient for the required biconditional. Thegeneral rule is

V=2d

[0142] where v is the number of vectors required to make the fix on theΦ-point, and d is the drop in the number of literals from the clauses ofthe given schema to the resulting clause in the target schema.

[0143] Another illustration of the Fix Rule is pqr v pq{overscore (r)} vp{overscore (q)}r v p{overscore (q)}r v {overscore (p)}qr v {overscore(p)}q{overscore (r)} v {overscore (p)}{overscore (q)}r, which is,however, equivalent merely to p v q v r, as it covers every line of thetruth-table except {overscore (p)}{overscore (q)}{overscore (r)}.

[0144] One pair of implicands is p{overscore (q)}r v pq{overscore (r)},which give the Φ-point p. But as this is a drop down from three lettersto one, we need a fix of four vectors or two vector sums on the point,and the third and fourth vectors {overscore (p)}qr and pq{overscore (r)}provide it. The same sort of fix appears with q(pqr v {overscore(p)}q{overscore (r)} and {overscore (p)}qr v pq{overscore (r)}) andr({overscore (p)}{overscore (q)}r v pqr and {overscore (p)}qr vp{overscore (q)}r).

[0145] A much simpler though negative example of the Fix Rule ispq{overscore (r)} v p{overscore (q)}{overscore (r)}, which seems to givep as an Φ-point resultant, but fails to for lack of a fix on the pointp, as four, not two vectors must converge on it for the drop. This actsas a constraint on the vector arithmetic. We seem to get p q r 1 1 1 + 1−1 −1 = 1 0 0

[0146] But the Fix Rule rules this out. If x columns are filled withnumbers, positive or negative, then the number of non-zero columns inthe sum must be x−1. The Fix Rule will seem entirely unartificial whenone recognizes that what it means in, say, a 3-space, is that a literalor one-letter proposition is a face, and so four corners are needed todetermine it. A two-letter proposition is a line, and so only twoletters are needed to fix it. And a point in a 3-space is a three-letterproposition.

[0147] Consider as another illustration of the Fix Rule pqrs vpqr{overscore (s)} v pq{overscore (r)}s v pq{overscore (r)}{overscore(s)} v p{overscore (q)}rs v p{overscore (q)}r{overscore (s)} vp{overscore (q)}{overscore (r)}s v p{overscore (q)}{overscore(r)}{overscore (s)}. This is very obviously equivalent to p, and sinced=3 for each clause, v for the point p=2d. So d=8, and eight vectors orfour vector sums are needed for the fix on the Φ-point.

[0148] In many cases the target schema is unbalanced in the sense thatits clauses have different numbers of conjuncts and so they need to beput into developed alternational normal form. An example pq vp{overscore (q)}{overscore (r)} v {overscore (p)}{overscore(q)}{overscore (r)} (Quine, 1982, p. 75). This is equivalent to pq v prv {overscore (p)}{overscore (q)}{overscore (r)}. Like the early Quine'sprocedure in “The Problem of Simplifying Truth Functions” (Quine, 1952,p. 524), the vector simplification method given so far has taken thecumbersome ‘developed normal formulas as the point of departure.’

[0149] If t is developed uniformly we get {overscore (p)}{overscore(q)}{overscore (r)} v p{overscore (q)}r v pqr v pq{overscore (r)} r,which in a V-diagram is clearly {overscore (p)}{overscore (q)}{overscore(r)} v pq v pr, as pr lies midway between pqr and 0, and pq lies midwaybetween pqr and pq{overscore (r)}. But without development, we can takethe ∴-point for p{overscore (q)}r v pq, which is pr, and argue thatsince pr→p{overscore (q)}r v pq (where “.expression” and “expression.”represents bracketing of the “expression” that precedes or follows thedots), and pr subsumes p{overscore (q)}r, the longer p{overscore (q)}rcan simply be replaced by its own implicant.

[0150] The equivalence of undeveloped

[0151] (i) {overscore (p)}q v p{overscore (q)} v {overscore (q)}r vq{overscore (r)}

[0152] and

[0153] (ii) p{overscore (q)} v {overscore (p)}r v q{overscore (r)}

[0154] is harder to establish. It is one which resists as many as twelvefell swoops (Quine, 1982, p. 76, also in 1952, pp. 523-527) or shortertruth-tables. In the vector space with developed alternational forms theequivalence is easy enough to see. The developed form of thisequivalence is easy enough to see. The developed form of this exampleis: {overscore (p)}qr v {overscore (p)}{overscore (q)}r v {overscore(p)}q{overscore (r)} v pq{overscore (r)} v p{overscore (q)}{overscore(r)} v p{overscore (q)}r.

[0155] The same result can be obtained using column matrices for thepairs of vectors. Then for {overscore (p)}qr v {overscore (p)}{overscore(q)}r we get p q r −1 1 1 + {overscore (p)}qr −1 −1 1 {overscore(p)}{overscore (q)}r −1 0 1 {overscore (p)}r

[0156] And for {overscore (p)}q{overscore (r)} v pq{overscore (r)} weget p q r −1 1 −1 + {overscore (p)}q{overscore (r)} 1 1 −1 pq{overscore(r)} = 0 1 −1 q{overscore (r)}

[0157] Similarly, for p {overscore (q)} r v p {overscore (q)} {overscore(r)} we get p q r 1 −1 1 + p{overscore (q)}r 1 −1 −1 p{overscore(q)}{overscore (r)} = 1 −1 0 p{overscore (q)}

[0158] It should be noted that in this example too the Fix Rule applies.It would be nice to take the vectors in a different order, so that p{overscore (q)}r and pq{overscore (r)} are chosen instead of p{overscore(q)}r and p{overscore (q)}{overscore (r)}, and also {overscore(p)}q{overscore (r)} and p{overscore (q)}{overscore (r)} instead of{overscore (p)}q{overscore (r)} and pq{overscore (r)}. This would yieldr and p instead of pq and q{overscore (r)} in the whole system. But thatwould mean dropping from three letters to one in the case of these twopairs of alternations, and we cannot do that as there is no fix on r oron p.

[0159] There may of course be more than one “shortest” schema. InQuine's example there is obviously is on inspection a second. The vectorsystem {overscore (q)}r v {overscore (p)}q v p{overscore (r)} has thesame overall “effect” in the vector-logical space.

[0160] Let us now try this example with the use of the

-points, the key prime implicants. Take first {overscore (p)}q v{overscore (q)}r. This alternation is implied by the

-vector which forms the parallelogram with 0. But there is no Φ-point,and so, apparently, {overscore (p)}q v {overscore (q)}r is notequivalent to {overscore (p)}r. Yet in the context of the whole schemep{overscore (q)} v {overscore (q)}r v {overscore (p)}q v q{overscore(r)}, it is. To see this, we move the free vectors p{overscore (q)} andq{overscore (r)} from the right-hand side of the V-diagram to theparallelogram on the left. The implicand of p{overscore (r)}, which isp{overscore (q)} v q{overscore (r)}, slides into place from p{overscore(q)} v q{overscore (r)} back to {overscore (p)}r, and the two-wayimplication or equivalence is established.

[0161] The vector summation of p{overscore (q)} and {overscore (q)}r top{overscore (qq)}r or p{overscore (q)}r is disallowed by the Fix Rule,according to which the number of vectors needed to make a fix is equalto the d-th power of 2. This summation would actually produce a negativevalue for d. As the number of literals rises from two to three, the dropincreases from 2 to 3, or −1.

[0162] Quine gives another interesting example of a simplification withfour simplest equivalents, one which also illustrates the method ofsimplification for non-developed or unbalanced schemata like the lastexample. The example (Quine, 1952, p. 528) is pqr v p{overscore (r)} vpq{overscore (s)} v {overscore (p)}r v {overscore (p)}{overscore(q)}{overscore (r)}{overscore (s)} (FIG. 23).

[0163] We begin by generating vector sums for the various disjuncts. Wecan see fairly easily that {overscore (p)}r and {overscore(p)}{overscore (q)}{overscore (r)}{overscore (s)} to start with, yield aparallelogram, but it seems to end at an

-point outside the logical space. Yet if we study that point, we can seethat it is actually at the co-ordinates {overscore (p)}{overscore(q)}{overscore (r)}{overscore (s)}{overscore (p)}r. This point, however,contains a contradictory or backward and forward instruction, namely ther from {overscore (p)}r and the {overscore (r)} from {overscore(p)}{overscore (q)}{overscore (r)}{overscore (s)} which can both bedeleted. There is also a double p in the final address, and one p ofthese, but not both, can of course be deleted. This leaves an end-pointfor a vector {overscore (p)}{overscore (q)}{overscore (s)}. By similarreasoning, we can arrive at the vector {overscore (q)}{overscore(r)}{overscore (s)} as the vector sum of p{overscore (r)} and {overscore(p)}{overscore (q)}{overscore (r)}{overscore (s)}. And similarly pqrwith p{overscore (r)} gives pq, or pqr with {overscore (p)}r gives the

-point qr. Each vector must be used at least once if it is not to appearunchanged in the simplified schema.

[0164] The corresponding Φ-points, however, do not appear at thedesignated addresses which are shortened versions of their

-points, and so the equivalence of the

-points and their implicands is not established. Just as in the exampleshown in FIG. 20, the drag-back effects described in connection with pand pp in FIG. 14 do not apply.

[0165] As before, in FIG. 24, we first write in the parallelogram from 0for the pair pqr v p r. This gives an

-point at pqp, and so from pqp we write in a pair of vectors to pqr andp r. Now pqr implies pq, and is subsumed by it, and we can represent thesubsumption rule here by drawing in the vector from pqr to pqp. FIG. 24is now showing an equivalence between pqr and pq, but only in thepresence of the p r in the alternational schema pqr v p r, i.e. with thetranslated or “borrowed” vector p{overscore (r)}→0.

[0166] It is worth realizing that in the reductions in developed normalform, the implicands can be replaced by the implicant only because ofthe various biconditionals or double arrows at work. There is nointrinsic magic in the Φ-point. In the undeveloped examples, too,clauses do not disappear in a general way because pairs of disjunctscollapse into their implicants, but because of the presence of specificconditions elsewhere in the schema, which translated have the effect orcreating biconditionals.

[0167] Finally, why is pq{overscore (s)} superfluous in the examplegiven in FIG. 23? The Answer is interesting and complicated, andprinciples about superfluity need to be established.

[0168] Take the truth, sometimes known as the Consensus Theorem, that pqv p{overscore (r)} v qr.

.pq v {overscore (p)}r. Representing this in the ANS-space for p, q andr, we can see that the implicant qr is the resultant of the disjunctionof pq and {overscore (p)}r (FIG. 25). We can give it as a general truththat implicants, in the ANS-space, are resultants.

[0169] Let the left-hand side of the Consensus Theorem be represented as

qp v {overscore (p)}r v qr

[0170] The Theorem says that the disjunct qr is superfluous. Considerthe dual of the left-hand side of the Theorem, in the CNS-space. It is

(q v p)({overscore (p)} v r)(q v r)

[0171] This is the conjunction ({overscore (q)} p)(p r)({overscore (q)}r). But clearly the last conjunct is superfluous, as the first twoconjuncts imply it by a hypothetical syllogism, in the sense that ifthey are true, so is it (FIG. 25).

[0172] It is nice to see the dual roles of conjunction and alternation,or ANS-and CNS-spaces, truth and falsehood, and how the concept of theresultant and the component binds them together.

[0173] In the following truth-table, we can that the (q r) resultant isso to speak “covered” by its component with respect to truth, in theANS-space, and falsity in the CNS-space. That is, with the disjunctionsin the ANS-space the addition of an extra truth on already true lines ofthe truth-table does not affect the truth of the whole schema. Andsimilarly in the CNS-space, if the whole schema is already false, addinga false conjunct will not affect that result. ANS- CNS- p q r qr v pq v{overscore (p)}r (q v p) (p v r) (q v r) T T T T T F T T F T T F T T F FF F F T T T T F T F F F T T F F F F F F

[0174] We are now in a position to deal with the superfluity of pqs inQuine's example in FIG. 24. Disjunctive clauses in the ANS-space likepq{overscore (s)} are superfluous when they are components. Before therepresentation of the schema pqr v {overscore (p)}{overscore(q)}{overscore (r)}{overscore (s)} v p{overscore (r)} v {overscore (p)}rv pq{overscore (s)} with a view to simplification, we can simply run acheck to see if any of the clauses are implicants or iota-points for anyothers. We can easily find that pq{overscore (s)}→.pqr v p{overscore(r)} from the truth-table for the schema; all the lines on whichpq{overscore (s)} is true are also lines on which either pqr orp{overscore (r)} is already true, and so pq{overscore (s)} can bedeleted from the schema to be simplified.

[0175] Geometrically, the construction is as follows: (FIG. 27). Notethat pq{overscore (s)} extends to pq{overscore (s)}p. This is howeverthe implicant for pqr{overscore (s)} v p{overscore (r)}. However,pqr{overscore (s)} itself extends to pqr{overscore (s)}pq. This lastschema is the implicant for pqr v pq{overscore (s)}. So pq{overscore(s)} gives way to p{overscore (r)} v pqr{overscore (s)}. Butpqr{overscore (s)} can itself can be dropped in favor of pqr vpqr{overscore (s)}. Any line of the truth-table for pqr on which is trueis also one which either pqr or p{overscore (r)} is already true. Hencepq{overscore (s)} can be dropped.

[0176] So for Quine's example in FIG. 23 we are left with the fourpossibilities:

[0177] These examples, and others like them, suggest the possibility offurther applications of simplifying geometrical theorems and methods tothe simplification problem.

[0178] The charm of a vector simplification technique is that is followsa least-action principle, for any number of propositional vectors, inthe sense that the problem is not one of finding shortest equivalents totruth-functional schemata. Rather the space, inasmuch as it is fixedvector space in which all free vectors having the same direction are ina sense the same directional vector, is unable not to give the desiredresult.

[0179] As to propositional logic as whole, it is nice to have all of thenineteen or however many clanking “rules of inference” within the space,so that there is just the one intuitively obvious method of argument:vector addition. It is really absurd to think of empty or “formal” rulessuch as association and communication as having the same status as saymodus tollens, which is genuine “motor” that advances arguments throughlogical space. Association and commutation should flow out of the natureof the logical space, and in the vector space they do. The vectors p v qand q v p, for example, have the same end-point, though they arrive atit by different but corresponding routes.

[0180] Part III

[0181] (i) Electrical and Integrated Circuit Minimization

[0182] Let us now see how the techniques described can be used in aroutine for simplifying electrical and integrated circuits. Take thetarget circuit ABC+A{overscore (C)}+AB{overscore (D)}+{overscore(A)}C+{overscore (A)}{overscore (B)}{overscore (C)}{overscore (D)} (FIG.28).

[0183] The first job is to plot this in the ANS-space as the set ofvectors pqr v p{overscore (r)} v pq{overscore (s)} v {overscore (p)}r v{overscore (p)}{overscore (q)}{overscore (r)}{overscore (s)}, as in FIG.23 above. Following the above-discussed general simplificationprocedure, in the ANS-space, we can simplify this system of vectors toe.g., pq v {overscore (p)}r v pr v {overscore (p)}{overscore(q)}{overscore (s)}. The resultant schema can then be translated intothe circuit diagram AB+A{overscore (C)}+{overscore (A)}C+{overscore(A)}{overscore (B)}{overscore (D)}, as illustrated in FIG. 29.

[0184] Note that the target circuit, as illustrated in FIG. 28, has fivegates (G=5), that the total number of inputs into these gates is twelve(I=12), and that the redundancy factor (i.e., the number of times anoriginal input is used again, corresponding to the join dots) is seven(R=7). These figures drop to G=4, I=9 and, most importantly, R=2 for thesimpler circuit, as illustrated in FIG. 29, representing correspondinggains in materials savings, speed and reliability.

[0185] (ii) Free Space Optical Computation

[0186] More than ten years ago the National Academy of Sciences Panel onPhotonics Science and Technology Assessment declared that ‘The ultimatebenefit of photonic processing could occur if practical optical logiccould be developed’ (Whinnery et. al., Photonics, 1988, p. 35). So farthe implied challenge of the Panel has not been met.

[0187] Vector manipulation has been one of the big success stories foroptical computation, but vector techniques themselves promise anapplication to the logic of optical computation as a whole. The fullANS-/CNS-space could be built as an optical device for checking thevalidity of arguments or as a logic device for optical computation, andalso as simplification machine. Each operation in the space is a laser,and the resultant proposition-points such as p and pq and pqr aremultifaceted beamsplitters or mirrors which reflect the beams in thecorrect logical directions at the correct logical strengths to ensurethe required implications.

[0188] Thus in FIG. 30 a beam V can be sent from the origin to ahalf-darkening beamsplitting mirror at the node p. At p it is split andsent at half-strength to q, and to {overscore (q)}. Simultaneously, asecond beam U from 0 is sent to the node {overscore (p)} v q, which isalso p→q. At this point U is split and sent at half-strength to{overscore (p)} and to q. The proposition p is said to “half-imply” q,in the sense that with one other proposition it does imply q, and theproposition {overscore (p)} v q is said to “half-imply” q in the sensethat with one other proposition (p) it does imply q.

[0189] Both half-implication beams are coincident on q, and at q thephotoreceptor gives a reading of 0.5+0.5 or 1. The system has opticallycomputed modus ponens; from an input of p and an input of {overscore(p)} v q, it has yielded up q. The system gives a physicalinterpretation of beamsplitting as multiple implication and of darkeningas fractional implication.

[0190] The same principles will apply to the other rules of inferenceand logical equivalences.

[0191] A development of the system given for modus ponens in FIG. 30obviates the need for a free beam for e.g., p to q, and simplifies thedesign of the node. In FIG. 31, the beam to p is split, atfull-strength, to p's implicants, which are p v q and {overscore (p)} vq (ignoring tautologies). The beamsplitter at {overscore (p)} v q itselfdirects the beam to q at only half-strength, and the desired computationis achieved.

[0192] We can also arrange that in an embodiment of the uninterpreted(p, q, . . . n) space, in which the base (p,q) is either p v q or pq(though not both), configurations of the beamsplitters will allow thenode to switch between the two states. A conjunctional state willcorrespond to a concave configuration, as exemplified in FIG. 32A. InFIG. 32A, both inputs are required for the activation of the node. Analternational state will correspond to a convex configuration, as shownin FIG. 32B.

[0193] (iii) “Flat” Optical Processing

[0194] An exemplary implementation of flat optical processing isillustrated in FIG. 34. Each position vector of cell 3400 is a lasersource, p, q, etc. An input beam 3404 is sent from p to a combiner 3410at O, and the same for a second beam 3402 from the q direction.Intensity filters 3406 and 3408 cut down the light from each source tohalf. The angle of the combiner 3410 is set so that both of the nowhalf-strength beams are coincident on the output in the (arbitrarilychosen)-p direction. At the output there is an optical filter 3412 witha sigmoid characteristic so that if the intensity of the beams is one orgreater (no greater than two), then the output is one, and if the inputis less than one then the output is zero, as illustrated in FIG. 35.Finally, at the output point there is a photoreceptor 3414, whichannounces that the two half-strength beams have converged on q for avalue of 1. This system will be optically on iff p and q are input.Since this implementation, acts as pq or an AND-gate, the output beamcan be used as a new p or input for subsequent computations.

[0195] So far what has been described is a single gate. Gates of thistype can be combined, however, using the sigmoid-characteristic filterto control the output. Take next an XOR-function representing thelogical schema p-q v −pq. In this case we can combine a p-q-gate with a−pq-gate, as in FIG. 36. Cell 3602 includes two input beams 3606 and3610 at p and −q respectively. Both input beams are passed throughrespective intensity filters 3614 and 3616 in order to attenuate thesignal in half. Both input beams are then combined at combiner 3620 anddirected to sigmoid filter 3628. Cell 3604 includes two input beams 3608and 3612 at −p and q respectively. Both input beams are passed throughrespective intensity filters 3618 and 3644 in order to attenuate thesignal in half Both input beams q and −p are then directed to sigmoidfilter 3626 by mirrors 3622 and 3624 respectively. The outputs from eachsigmoid filter 3628 and 3626 pass through respective intensity filters3642 and 3640 respectively in order to attenuate the signals in half.The attenuated beams are then directed to sigmoid filter 3636 viamirrors 3630, 3632, and 3634. The output of sigmoid filter 3636 isdetected by photodetector 3638. These gates can of course be furthercombined for example to yield an optical analogs of other integratedcircuits, e.g. IC 74266, which is entirely composed of XOR-gates.

[0196] The XNOR-gate can be similarly represented, for example as acombination of a pq-gate and a −p-q-gate as illustrating in FIG. 37.Cell 3702 includes two input beams 3706 and 3708 at q and prespectively. Both input beams are passed through respective intensityfilters 3714 and 3716 in order to attenuate the signal in half. Bothinput beams are then combined at combiner 3722 and directed to sigmoidfilter 3628. Cell 3704 includes two input beams 3710 and 3712 at −q and−p respectively. Both input beams are passed through respectiveintensity filters 3718 and 3720 in order to attenuate the signal inhalf. Both input beams −q and −p are then directed to sigmoid filter3626 by mirrors 3726 and 3724 respectively. The outputs from eachsigmoid filter 3628 and 3626 pass through respective intensity filters3642 and 3640 respectively in order to attenuate the signals in half.The attenuated beams are then directed to sigmoid filter 3636 viamirrors 3630, 3632, and 3634. The output of sigmoid filter 3636 isdetected by photodetector 3638.

[0197] The NAND- or −(pq)-gate is more complicated in an interestingway. It consists of the three cells that represent those three lines ofthe truth-table which negate the pq line. In the vector representation−(pq) or (by De Morgan's Theorem −p v-q) is given as illustrated in FIG.38.

[0198] The optical implementation of −p v-q is illustrated in FIG. 39,the NAND-gate, which consists of three cells plus filters. Cell 3902includes two input beams 3908 and 3910 at p and −q respectively. Bothinput beams are passed through respective intensity filters 3916 and3918 in order to attenuate the signal in half. Both input beams are thencombined at combiner 3928 and directed to sigmoid filter 3938. Cell 3904includes two input beams 3912 and 3913 at −p and q respectively. Bothinput beams are passed through respective intensity filters 3922 and3920 in order to attenuate the signal in half. Both input beams q and −pare then directed to sigmoid filter 3940 by mirrors 3930 and 3932respectively. Cell 3906 includes two input beams 3914 and 3916 at −p and−q respectively. Both input beams are passed through respectiveintensity filters 3926 and 3924 in order to attenuate the signal inhalf. Both input beams −q and −p are then directed to sigmoid filter3942 by mirrors 3936 and 3934 respectively. The outputs from eachsigmoid filter 3938, 3940, and 3942 pass through respective intensityfilters 3954, 3952, and 3640 respectively in order to attenuate thesignals by ⅓. The attenuated beams are then directed to sigmoid filter3948 via mirrors 3950, 3944, 3946 and 3952. The output of sigmoid filter3948 is detected by photodetector 3638.

[0199] It is also easy to use the same techniques in the SOP-orConjunctional Normal Form. Modus Ponens states that if ‘If p then q’ and‘p’ then ‘q’. As illustrated in FIG. 40, a CNS-cell will produce theoutput q if the inputs are p v −q and p. Cell 4002 includes two inputbeams 4004 and 4006 at p and p v −q respectively. Both input beams arepassed through respective intensity filters 4018 and 4020 in order toattenuate the signal in half. Both input p is then directed to sigmoidfilter 4014 by mirror 4008, whereas input p v −q is directed to sigmoidfilter 4014 by mirror 4010 and combiner 4012. The output of sigmoidfilter 4014 is detected by photodetector 4016. As illustrated in FIG.41, for p→q has as translated or logically equivalent forms O→−p v q andp v −q→O.

[0200] A second method of exploiting the vector system for computationis more markedly spatial. Represent the propositions in theuninterpreted (p,q) space with spatial light modifiers (SLMs). When thefirst premise is input, e.g. {overscore (p)} v q, then the origin 0, andwith it the position of the whole space, are moved to the point{overscore (p)} v q, or in a {overscore (p)} v q direction. We could saythe 0 becomes {overscore (p)} v q, so that we are now in a {overscore(p)} v q environment, a {overscore (p)} v q world. Then p in the secondSLM (FIG. 33) will be q, and we have modus ponens. And when the wholespace is displaced in a {overscore (p)} v q direction, q is p!

[0201] A second method of exploiting the vector logical characteristicsof light is more markedly spatial. The premises or inputs can berepresented in the CNS-space as spatial light modifiers. For example,let there be two modifiers 4202 and 4204, which represent the wholeCNS-space for two variables p and q as illustrated in FIG. 42A. Theorigin of the first modifier 4202, representing −p v q, is aligned withthe second modifier 4204, so that the origin 4202 is at −p v q on 4204,as illustrated in FIG. 42B. This represents a −p v q shift within 4204.We can say that if O becomes −p v q, then p becomes q, or even that whenwe are in a −p v q environment p is q. FIG. 43 shows an oblique view ofthe two SLMs in sequence in combination with a light source 4302,whereas FIG. 44 shows a side view. As illustrated in FIG. 43, lightsource 4302 incident on p on first modifier 4202 yielding q on thesecond modifier 4204. It is as if we are asking: if O is −p v q, what isp? The answer computed is: q. The results of the vector logic system forpropositional logic show that this implementation technique can be usedfor all the rules of argument, and for any number of variables.

[0202] (iv) Colorimetric Processing

[0203] Colored laser beams can be used so that the refractive angle isbuilt into the vector rather than into the propositional nodes as theCIE (Commission Internationale de l'Eclairage) x-y chromaticity diagram(a color mixing diagram) is itself a vector space. (Further a mixedsystem of colored laser and colored mirrors could be used. Opticalcomputation for simplification may then merely include the colorimetricprocess of additive color mixing. For example, in the CNS-space, let pbe red (R), {overscore (p)} a complementary cyan blue-green (C), q ayellow (Y), {overscore (q)} a complementary blue (B), p v q yellow-red(YR) and {overscore (p)} v q blue-red (BR). Also {overscore (p)} v{overscore (q)} is the complementary of YR, a cyan blue.

[0204] The contradiction of 0 (the so-called “Nullpunkt”, or “white”)corresponds to the addition of complementary hues. For example, YR+BR=R,since Y and B are complementary.

[0205] With these calorimetric assignments we can compute modus ponensand the other rules of argument and truth-preserving substitutions. Forexample, as illustrated in FIG. 45, if proposition p is R from a lightsource 4504 that passes through a red filter 4508, and if proposition{overscore (p)} v q is CY from a light source 4502 that passes through aCY filter 4506, and if q is Y, from colorimiter 4510, then p→q, or{overscore (p)} v q, is CY. Together with R this give Y or q, as C and Rare complementaries.

[0206] In the ANS-space we can perform simplifications colorimetrically.For example, in FIG. 46, exemplifies a basic simplification in which pqv p{overscore (q)} is equivalent to p. Let YR, from yellow-red filter4606, represent pq, and BR, from blue-red filter 4608, representp{overscore (q)}. The Y and B portions of the beams cancel atcolorimiter 4610 because they are compliments, leaving RR or R, which ispp or p.

[0207] The CIE xY chromaticity chart, for example as illustrated in FIG.47, is a vector space. If colored beams are used for V, U, W . . . N,the rules of vector addition, subtraction and displacement in the colorspace represent the CNS or ANS-space. Logic operations then become therules of colorimetry.

[0208] For example, let p be red (R), say 620 nm., p a complementarycyan (C) at 494 nm., q a yellow (Y) at 575 nm., and q a blue (B),complementary to Y, at 470 run. Then, working in the CNS-space withthese hue assignments, we can compute modus ponens colorimetrically, forexample as illustrated in FIG. 48. The addition of p or R, for example620 run., and −p v q or CY, for example having a dominant wavelengtharound 530 nm., with colorimeter 4802 is the yellow Y of 575 nm.: q. The“tautology” at white (W) is the addition of complementary hues.

[0209] In the ANS-space black is the contradiction. With complementariesin the ANS-space‘. . . what is offered, so to speak, by one [reflection]spectrum (or colour) is withdrawn by the other, so that the result is avanishing of colour, just as in the contradiction between twopropositions which negate one another the result is a vanishing ofinformation’ (Jonathan Westphal, Colour, Oxford, Blackwell, 2nd. ed.,1991, p. 108).

[0210] In the ANS-space we can perform exactly parallel computationscolorimetrically, in particular simplification routines. Take the mostbasic simplification as an example as illustrated in FIG. 49, whereinthe colorimetric cell will yield a signal of a specified output, say forpq v −p−q. Let p be R, for example 620 nm., let q be Y, for example 575nm., let −p be C, for example 494 nm., and let −q be B, for example 485nm. The input p is reflected off mirror 4902 to be combined with theinput q with combiner 4904. The input −q is reflected off mirror 4910 tobe combined with the input −p with combiner 4912. The combination pq isreflected off mirror 4906 to be combined with the combination −p−q withcombiner 4908. The calorimeter 4914 provides an output of p, or in otherwords R, from input pq v −p−q. The analog of the sigmoid filter in thisparticular colorimetric application of vector logic is the color filterwhose ideal transmission curves are shown in FIG. 50, with90%+transmission peaks at 485 nm. and 595 nm. It is an OR-gate; there isa signal through the filter iff either p and q or −p and −q. As such,the schema pq v −p−q is logically equivalent to p. YR will represent pq,and BR is pq. The Y and B components cancel, leaving RR or R, which isp.

[0211] (v) Polarization Based Processing

[0212] The vector logic system also provides for optical AND, NAND, NOT,and the other logical functions, implemented in optical gates in whichthe input and output are coded directionally in a more purelygeometrical form. The implementation does not call for a nonlinearoptical material, but instead embodies the vector logical analysis ofthe AND-function in its developed CNS-form, and well-known opticalmaterials: reflectors or mirrors, beamsplitters, retarders andpolarizers.

[0213] Each gate, or cell, may comprise of a group of optical elementsarranged in three layers. The elements are reflectors, polarizers andretarders. The three layers in each cell represent the conjunctionalnormal form (CNS-plane), the alternational normal form (ANS), and anintermediate transformational layer (T-plane). The input to each cell isthe optical beam (or beams) which enter the cell, from any inputdirection. The output from each cell is the optical beam (or beams)which leave the cell.

[0214]FIG. 51 illustrates an exemplary AND-cell 5100 in accordance withone embodiment of the present invention. Consider the input of p 5108and q 5110 into the top CNS-plane 5102. A p v q input would occur onlyon the ANS-plane. The two entry vectors, p and q travel towards thecenter or origin O-CNS of the cell. At 0 the p-beam 5108 is split bybeam splitter 5112 into two optical components, traveling to the pointsp v q and p v −q. This is the optical analog of the developed form of p.Pairs of reflectors and polarizers are placed at the corners of thecell, as well as a 180 mμ phase-shifter or retarder in two of theadjacent corners. The q-beam is also split into its developed form bythe beam-splitters at O, and directed to −p v q and to p v q.

[0215] At this point the three resultant beams (−p v q, p v q, and p v−q) are directed to the center of the T-plane. When the two polarizedbeams −p v q and p v −q meet at the origin O₁T they are extinguished, asthey are out of phase by 180°. If the distances from 0₁T to the verticesof the cell are correctly set, then there will be a local null-value forthe output at the points at which the exit vectors emerge from the cell,i.e. at p v −q and −p v q, as well as at O₁T. The T-plane also has asecond sub-plane on which the p v q and −p v −q beams would meet, with acenter at O₂T. But in the case of the three beams from p and q inputs,there is no −p v −q beam, and the p v q beam, unextinguished, travelsonto the ANS-plane. It continues as the exit vector or output of theANS-plane in a pq direction: it has become the output pq. At this pointthe output pq can be reflected as new input to the next cell, and thefurther outputs arranged in a cascade of cells with the different logicfunctions. But if the output is negative, it must be entered asnegative, −p or −q.

[0216] If for example a combination of inputs is given, which is part of−(pq), i.e. any of −pq or p-q or −p-q, then the exit vectors will be oneor more of these conjunctions on the ANS-plane, and this will be routedas negative output. When such a negative output is fed into a NOT-cell5204, which has the function of inverting all inputs by reflection androuting them to the dual plane, it is output as a positive signal, asillustrated in FIG. 52.

[0217] Specifically, when the output of the AND-cell 5202 is input in tothe NOT-cell 5204, the final output is −p if pq is input. If p and q areinput, −(pq) is “logically” off though optically on as the −p exitvector and p entry vector in the next cell. It is an important featureboth of the vector logic system itself given in “Logic as a VectorSystem” and of the present directional implementation that a negativelogic output, −p, say, is optically positive, just as −p is vectoriallypositive in the sense that there is a directed line segment −p, whichhappens to point in a reverse p direction. If −p is true then the −pbeam is on, though logically negative, as all left-hand or down-tendingbeams in the CNS-plane are coded negative. An important related featureof the implementation is that input beams are to be set in defaultnegative states such as −p, −q, etc., for the first-level inputs, sothat at the onset of computation the inputs to the system are logicallyoff though optically on.

[0218] Consider the implementation of the COIN-function or coincidencefunction, in the cells shown in FIG. 53. The initial off-inputs are −pand −q in both AND-cells 5302 and 5304. Without on-input, the −p−q-cell5302 gives an output to the OR-cell, while the pq-cell 5304 does not.What happens when the overall input is say p-q? In both the AND-cells at5302 and 5304, there is no positive output, and hence no positive outputin this case for the pq v −p−q cell. The output in the pq-cell isnegative, and hence fed to −p in the OR-cell 5306. So the OR-cell 5306registers a NOT-state if p-q is input into the AND-cells 5302 and 5304.Only at the final output or outputs should the values of the logic andoptical positive and negative be caused to coincide, and negativesignals be extinguished.

[0219] Precisely the same principles apply in the implementation of theXOR-function, or half-adder FIG. 54 and to the MUX-function ormultiplexer FIG. 55.

[0220] Finally, to illustrate how cascades of more than one level can beimplemented in the optical logic described here, consider how theNAND-function provides a way of saying pq. The following is equivalentto pq: (pNANDq)NAND(pNANDq). What we have to do to create an opticalcircuit which expresses this function is to build a sequence of twoNAND-cells conjoined into another NAND-cell. FIG. 56 shows this sequencefor an input of p and q, FIG. 57 for an input of p and −q. This examplealso shows, of course, that any logical function and any opticalintegrated circuit can be implemented within the optical logic which hasbeen described.

[0221] It is worth emphasizing again that these gates and circuits inthe fourth embodiment use only four principles of implementation:beamsplitting; reflection; polarization; and retardation. These areimplementations which can be scaled down as far as desirable. There isno scale at which reflection of a stream of light cannot take place, andeven at the smallest level photons can be polarized simply by areflecting surface, as, for example, on a highly polished metallicsurface of an automobile. As for retardation, there is a sense in whichit does not even need a physical embodiment within the system, as it canbe organized by the architecture of the gates involving different pathlengths of the corresponding beams.

[0222]FIG. 58 depicts an exemplary embodiment of a NOR-gate, orSheffer-function −(p v q), in accordance with the present invention. Inthe cell 5802 at the origin, four beam-splitters 5820 break up inputfrom p, q, {overscore (p)} and {overscore (q)} into vectors exiting asthe developed form, so that p, for example, becomes p{overscore (q)} v.pq. The exit vectors from the cell are the normal of the (p, q) pointcoordinates, for example exits at {overscore (p)} {overscore (q)}. Exitvector e point positions: {overscore (p)}{overscore (q)} 7:30 {overscore(p)}q 4:30 pq 11:30  pq 1:30

[0223] The exit vector for {right arrow over (pq)} is blocked in thenor-cell, represented by the shaded port at {overscore (p)}{overscore(q)}.

[0224] (Two orthogonally oriented polarizers are present at the entry tothe ports {overscore (p)}q and p{overscore (q)}

[0225] Suppose now inputs in the following pairs of literals: a) pq b)p{overscore (q)} c) {overscore (p)}q d) {overscore (p)}{overscore (q)}

[0226] These inputs will explain further features of the cell as well asits overall “

” function and output. For example as illustrated in FIG. 59.

[0227] The input beams {overscore (p)} and {overscore (q)} are split atO to p 1 v pq and pq v {overscore (p)}q respectively. This gives adouble beam to p 1, which is Π 1. But the p{overscore (q)} v pq outputport is blocked. At the p{overscore (q)} and {overscore (p)}q outputsthe two beams are orthogonally polarized, and they are then directedtogether at E_(x), the xnor sum at the top right of the cell. (Theactual spatial location) of these last ports E_(x), and E_(f) isarbitrary.). As there is no output at pq, the final output, herelabelled E_(f), is zero, wherein there is no light, and (p:1, q:1=0).

[0228] b) p{overscore (q)}—Take next the input (p, 1), as in FIG. 60.

[0229] The input beams {overscore (p)} and {overscore (q)} are split atO to pq v. p{overscore (q)} and p{overscore (q)} v {overscore(p)}{overscore (q)} respectively. The pq output is blocked as before,but it is only 25% of the total output, which is itself twice the input.The p{overscore (q)} {right arrow over (out)}put is polarized at zero°,but there is no p{overscore (q)} {right arrow over (outp)}ut, so thatE_(x) is on, and since {overscore (p)}q so is E_(f), consequently (p=1,1:1,=1). So far we have: pq pq; 1 1 0 1 0

[0230] c) Πq

[0231] In FIG. 61, the input is ({overscore (p)}q). The input beams{right arrow over (p q)} are split at O to give developed {right arrowover (pq v. pq)} and {right arrow over (pq v. pq)} respectively. Thedouble {overscore (p)}q beam is polarized and reflected to E_(x) andthence to E_(f), as there is no orthogonally polarized {right arrow over(p q)} output at {overscore (p)}q. The {right arrow over (pq)} output isblocked, as before, but E_(f)=1, even without the contribution of {rightarrow over (p q)}

[0232] We now have for the cell at E_(f): pq pq; 1 1 0 1 0 1 0 1 1

[0233] d) Turning finally to input ({overscore (p)} {overscore (q)}), asin FIG. 62, the logical pattern is similar. The E_(x) or XNOR-functionis extingquished, but the double {right arrow over (p q)} componentbrings the sum to 1 at E_(f). The block at ({overscore (p)} {overscore(q)}) is unnecessary as there is no {right arrow over (pq)} componentsat this exit port.

[0234] The result overall is p

q for the cell is: pq pq 11 0 10 1 01 1 00 1

[0235] Furthermore, exploiting the geometrical nature of the light beamsin a matrix around the origin rather than in an optically bistablephysical medium, is conceived as functioning like a silicon plate in asemiconductor chip.

[0236] The basic cell concept here described is complete, and can byitself be used to construct cascades of other functions in a gatesequence. But in addition, we can exploit the concepts to devise anyother logic function, most importantly p

q the NAND or Pierce function in the dual CNS-space or vector sub-space.These cells are all-optical and can be put together to implement thelogic functions of decoders, multiplexers, adders, and the rest,including of course, router switches.

[0237] (vi) Polarization Based Processing With aDualization/Polarization Corridor There is another way to implement thevector logic system using polarization which does away with thepotential traffic jam at the center of the AND-cell. Let us take onceagain AND-function, and consider two inputs, p and q, which enter theAND-cell shown in FIG. 63.

[0238] At each literal (p, −p, q, −q) of the input plane cell there is abeam-splitter will have the effect of dividing the p and q beams intotwo beams each. These two beams will represent p v q and p v−q, for p,and p v q and −p v q, for q. The logical equivalent of “p and q” usingthe “or” function is the so-called developed conjunctional normal form“(p v q) and (p v −q) and (−p v q)”, as can be seen from the truth-tablein FIG. 64. What has happened is that optically, using the vector logicrepresentation, we have modeled this form. The same would happen withother literal inputs −p and −q.

[0239] The next step is to realize something interesting and importantabout the developed conjunctional normal form. If we take the firstconjunct, in this case p v q, we can arrive at the proposition “p and q”by changing the “or” to an “and”, and dropping the other two conjuncts.But what we can notice here about the other two conjuncts is that theyare opposed vectors in the vector logic space: they are p v −q and −p vq. This rule applies no matter how large the conjunction is. Say it's −pand q and r″, an 8-line table. Then we can take the rather largedeveloped conjunctional normal form, note that the conjunction has T onevery line of the truth-table except the last, and that the remaining 2to the Nth minus 2 rows (6 of them) are opposed in vector space.Examining the truth-table in Diagram 2, we can see that any conjunctionof no matter what length can be expressed by deleting the vector opposed(or technically S-opposed) rows, in this case Rows 2 and 3), andconverting the main operator (or) into its dual. It is as if one hadsaid that if the S-opposed values are deleted, then the derived valuesfor pq and p v q are the same, and p v q is pq!

[0240] An optical analog of this is a polarization cancellation of onebeam by another. Imagine that the three beam groups we have in the cellon the CNS-plane ((p v q)(p v −q)(−p v q)) are now reflected and, at theentrance to the polarization corridor, polarized orthogonally withrespect to one another, in such a way that the opposed vectors in theCNS-plane cancel, for example p v q and −p v −q, or p v −q and −p v q.We have the extinction at a point at which the two “logically opposed”beams are incident. We can tag the exit vector at 1:30 in the exit planein Diagram 1 pg, and we can enter it into other cells as an input, p orq or whatever. Thus we have a cascade, as before in (iv), and thegeneral structures of these cascaded cells are the same.

[0241]FIG. 63 also shows how the beams in the dualization orpolarization corridor are to be managed to as to drop Operator S-opposeddisjuncts of conjunctions. (Conjunctions of disjunctions with oppositeliteral values are Operator S-opposed, disjunctions of conjunction areOperator N-opposed.). When p and q are entered on the input plane, inthe lower four lines of polarization corridor (representing theXOR-function) we get p from p v −q and −p from −p v q. But these areorthogonally polarized and cancel. In the upper four lines of thepolarization corridor representing pq and −p-q, the so-calledCOIN-function for (COINcidence), the p and q beams have the same planeof polarization and so do not cancel one another. The polarizationsgoverning the XOR-function in the bottom for lines of the dualizationcorridor ensure that the extraneous non-AND elements of the conjunctivenormal form expansion for any coincident conjunction are extinguished,and the COINfunction dos the same for the non-coincident conjunctionssuch as p and −q.

[0242]FIG. 65 shows an all-optical data distributor, decoder or DMUX(logical demultiplexer), with a message r being delivered to Alice, ataddress pq, or to Bob, at p-q.

[0243]FIG. 66 shows a latching function for a simple set-reset flip-flopfor all the possible optical paths. Here it is important to note thateven when the unit is logically off, optically it is on, and this is howthe stored “set” value Q is held. In this way optical storage isachieved using the AND-function within the NOR-function version of theSR flip-flop.

[0244]FIG. 67 shows the same flip-flop for values S=1, R=0, present Q=1,for a new Q or output of 1.

[0245] In conclusion, consider four interesting features of the opticalsystem. Interesting Feature A is that in the vector logic system thereis a sense in which reverse vectors are both positive. In theimplementation we see that both p and −p for example are equally realoptically. So logic negative is not optical negative. Of course at theend of the desired computation, the SR flip-flop-function for example,we take account of the positive output, and ignore the negative output.At this point optical positive and logic positive will coincide. Thisfeature is important for the flip-flop, as the light for the latchedvalue comes from the optically positive signal even when that value isnegative.

[0246] Interesting Feature B is connected with this. It is that whendata is entered into the system, say p and −q, then that will be amatter of killing off the −p beam and killing off the q-beam. We mustimagine that initially all the beams are optically on, and that weselect our input by removing the beam in the opposite direction. Or ifyou like we can imagine that at the outset the default state is one inwhich the −p beam and the −q beam are on, and that they stay on unlessredirected.

[0247] Interesting Feature C is that for any function built out of theAND and the NAND-functions if the input power of the variables such as pand q is 1, then the output is 1, leaving aside operating losses fromimperfect polarization extinction ratios, losses on reflection and soforth.

[0248] Interesting Feature D is perhaps the most important. Thepolarization based-implementations are clearly reversible, in the sensethat no information is destroyed anywhere in the system. What the vectorlogic implementation does is to move information around, withoutdestroying any physically, and to selectively use the output. In thecase of a NAND-cell, for example, the negative outputs are fed asparallel beams into the positive input of the next cell. They are asclose together as is necessary for the next stage of processing, butthey are distinct beams. Any input into a cascade is recoverable fromthe output, and information is conserved.

[0249] Can the implementation of the present system be reconciled withthe character of the information in the bitstream emerging from a fiberoptic cable? That is, is an OEO-conversion required before informationin the bitstream can undergo logical processing with the vector logicsystem in the implementations described in (iv) and here in (v)? Theanswer is that the mode of the signal of the fiber can be made to encodespatial information, at least over relatively short distances, evenafter enormous twisting and even knotting of the cable. An initialcalibration can be used to determine the exit vectors of the p and qvariables at the end of the fiber.

[0250]FIG. 68 shows the coding of p, q, −p and −q in the propagation oflight rays through a short piece of fiber. For the correct length theinput will match the output. So the with this method of coding thespatially or vector coded information can be introduced directly into aprocessing unit composed of the optical AND, NAND and other gatesdescribed.

[0251]FIG. 69 illustrates an optical AND gate and switching device usingpolarization photochromism. The AND-gate described in FIG. 69 is aswitch having an optical output in a selected direction of I iff (if andonly if) the inputs from the incident beams, here called the p- andq-beams, are both 1. The pq-beam (p and q) is on iff the p-beam is onand the q-beam is on. This is achieved by using: a polarizationphotochromic with a given plane of polarization as the so-called“switching medium”; a q-beam has a control signal which shifts thepolarization plane of the switching medium ninety degrees; and a p-beamwhose plane of polarization is orthogonal to the original plane ofpolarization of the switching medium. When the q-beam is off, the p-beamsuffers extinction in the switching medium. But the q-beam is directedat an angle which is not the output direction. The result is a genuineoptical AND-gate which can be used in switching and processingapplications.

[0252] The AND-function is basic for the other more complex logicalfunctions necessary in high-speed photonic signal systems intelecommunications and computational applications generally. This hasbeen the source of the great interest taken recently in opticallybistable devices and materials. Yet optical materials seem intrinsicallyunsuited to the production of bistable state, unlike their analogs insemiconductor technology. However, the polarization photochromics are anexception, as they are materials which do exhibit the demandcharacteristics, in the given configuration. These characteristics aregiven in the truth-table: pq pq 11 1 10 0 01 0 00 0

[0253] More complex optical switches such as multiplexers anddemultiplexers, which are themselves logical functions of input orswitches can therefore be constructed. It is commonly recognized thatthe changes in the absorption profiles of photochromic polymersgenerally have optoelectronic, optical storage and logic-switchingapplications, but the uniqueness of the present invention is that itshows how the latter can be made specifically with the polarizingphotochromics.

[0254] The heart of the optical AND-gate or switch described here (FIG.1), which is entirely novel and for which there is no prior art, is aswitching medium fabricated of a material such as but not limited to theliquid crystalline line polymers listed by Ichimura, described in thebackground of the invention, above, which display the requiredphotochromism with respect to polarization

[0255] In FIG. 69, the p-beam is directed towards the output through theswitching medium, and is in a polarization state orthogonal to that ofthe polymer. It therefore suffers extinction at the surface of thepolymer unit.

[0256] The q-beam acts as an angle to the p-beam. It consists ofpolarized actinic or UV light. It therefore induces a 90-degreephotochromic change in the angle of polarization or polarization sate ofthe switching medium. The q-beam is arranged to strike the polymer cellbefore the p-beam e.g. lengthening the path of the q-beam relative tothe p-beam. It reorients the plane of polarization of the polymer unitprior to the arrival of the p-beam. The result is that the unit nowblocks the p-beam.

[0257] It is still true, as Norbert Streibl et. al. (“Digital Optics”,(1989)) pointed out, the ‘A uniform technology for digital opticalinformation processing, comparable in its significance tomicroelectronics, does not yet exist and is by itself a challengingresearch goal.’ A vector logic for optics is a source from which such a“uniform” technology can flow, just as electronics derived from thenatural isomorphism of electric circuitry and truth-functional logic.

[0258] In this disclosure, there is shown and described only thepreferred embodiment of the invention, but, as aforementioned, it is tobe understood that the invention is capable of use in various othercombinations and environments and is capable of changes or modificationswithin the scope of the inventive concept as expressed herein.

[0259] Although certain specific embodiments of the present inventionhave been disclosed, it is noted that the present invention may beembodied in other forms without departing from the spirit or essentialcharacteristics thereof. The present embodiments are therefore to beconsidered in all respects as illustrative and not restrictive, thescope of the invention being indicated by the appended claims, and allchanges that come within the meaning and range of equivalency of theclaims are therefore intended to be embraced therein.

What is claimed is:
 1. An optical AND-gate comprising: first and secondoptical inputs; an optical output; first and second optical pathsleading from said first and second respective optical inputs to saidoptical output; intensity filters located within each of said first andsecond optical paths; and an optical filter having a sigmoidcharacteristic located at a position common to both said first and saidsecond optical paths.
 2. An optical processor comprising: the opticalAND-gate of claim 1; two light sources providing optical signals intosaid first optical input and said second optical input respectively; anda photodetector for detecting said output of said optical filter havinga sigmoid characteristic.
 3. An optical XOR-gate comprising: first,second, third, and fourth optical inputs; first, second, third, andfourth optical outputs; first and second optical paths leading from saidfirst and second respective optical inputs to said first optical output;third and fourth optical paths leading from said third and fourthrespective optical inputs to said second optical output; intensityfilters located within each of said first, second, third, and fourthoptical paths; a first optical filter having a sigmoid characteristiclocated at a position common to both said first and said second opticalpaths; a second optical filter having a sigmoid characteristic locatedat a position common to both said third and said fourth optical paths,wherein said third and fourth optical outputs from said respective firstand second optical filters having a sigmoid characteristic being incommunication with a third optical filter having a sigmoidcharacteristic.
 4. An optical XNOR-gate comprising: first, second,third, and fourth optical inputs; first, second, third, and fourthoptical outputs; first and second optical paths leading from said firstand second respective optical inputs to said first optical output; thirdand fourth optical paths leading from said third and fourth respectiveoptical inputs to said second optical output; intensity filters locatedwithin each of said first, second, third, and fourth optical paths; afirst optical filter having a sigmoid characteristic located at aposition common to both said first and said second optical paths; and asecond optical filter having a sigmoid characteristic located at aposition common to both said third and said fourth optical paths;wherein said third and fourth optical outputs from said respective firstand second optical filters having a sigmoid characteristic being incommunication with a third optical filter having a sigmoidcharacteristic.
 5. An optical NAND-gate comprising: first, second,third, fourth, fifth, and sixth optical inputs; first, second, third,fourth, fifth, and sixth optical outputs; first and second optical pathsleading from said first and second respective optical inputs to saidfirst optical output; third and fourth optical paths leading from saidthird and fourth respective optical inputs to said second opticaloutput; fifth and sixth optical paths leading from said fifth and sixthrespective optical inputs to said third optical output; intensityfilters located within each of said first, second, third, fourth, fifth,and sixth optical paths; a first optical filter having a sigmoidcharacteristic located at a position common to both said first and saidsecond optical paths; a second optical filter having a sigmoidcharacteristic located at a position common to both said third and saidfourth optical paths; and a third optical filter having a sigmoidcharacteristic located at a position common to both said fifth and saidsixth optical paths; wherein said fourth, fifth, and sixth opticaloutputs from said respective first, second, and third optical filtershaving a sigmoid characteristic being in communication with a fourthoptical filter having a sigmoid characteristic.
 6. A colorimetricoptical processor comprising: a first light source for providing a firstoptical input; a second light source for providing a second opticalinput; and first and second optical paths leading from said first andsecond respective optical inputs to a photodetector; wherein saidphotodetector detects color of light from said first light source andsaid color of light from said second light source and outputs an outputcolor based on additive color mixing of color of said first light sourceand color of said second light source and based on a x-y chromaticitydiagram.
 7. A polarization based optical processor comprising: a firstlayer representing the conjunctional normal schemata plane; a secondlayer representing the alternational normal schemata plane; and anintermediate transformational layer.
 8. A polarization based opticalprocessor of claim 7, further comprising: a beam splitter in said firstlayer for splitting an input beam into two orthogonal beams; reflectorsin said first layer for reflecting said two orthogonal beams to saidintermediate transformational layer; a first polarization reflector insaid second layer for reflecting one of said two orthogonal beams fromsaid first layer into a direction parallel with said intermediatetransformational layer; a second polarization reflector and retarder insaid intermediate transformational layer for reflecting with a phasedelay, said other one of said two orthogonal beams from said first layerinto a direction parallel with said intermediate transformational layer;a combiner for combining said beam from said first polarizationreflector and said beam from said second polarization reflector, and fordirecting said combined beams toward a third reflector, said thirdreflector positioned to direct said combined beams to said second layer.9. An optical AND gate, comprising: a. a photochromic polarizer; b. aplane polarized beam of light, incident on said polarizer; and c. acontrol beam of actinic ultraviolet light for controlling the state ofthe polarizer to selectively block or pass the beam of light incident onsaid polarizer.